Table 12.14. The four data sets. The second column gives the latitude and longitude coordinates in degrees for the upper-left and lower-right corners
of the terrain. The south and west coordinates are negative.
coordinates grid size
filt. length simplices
Sine na
100 × 100
10,001 59,996
Iran 42, 42, 23, 65
277 × 229
63,434 380,594
Himalayas 46, 66, 24, 105
469 × 265
124,286 745,706
Andes 15, –87, –58, –55
385 × 877
337,646 2,025,866
North America 55, –127, 13, –61
793 × 505
400,466 2,402,786
implemented the algorithms for constructing QMS complexes and computing the persistence of the critical points. My implementation for the former uses a
different algorithm than the one presented in this chapter. The algorithm uses edge tags to reroute paths using a single pass through the critical points.
12.5.1 Data
We use four rectangle sections of rectilinear 5-minute gridded elevation data of Earth National Geophysical Data Center, 1988 and one synthetic data sam-
pled from h x, y = sin x + sin y for input. Table 12.14 gives the names and
sizes of the data sets. Each data set is a height function h : Z
2
→ R, assign- ing a height value h
x, y to each point of its domain. Consequently, we may view the data sets as gray-scale images, mapping heights to pixel intensities,
as in Figure 12.9. In each case, we compactify the domain of the function, a gridded rectangle, into a sphere by adding a dummy vertex at height minus in-
finity. We then triangulate the resulting mesh by adding diagonals to the square cells. As a result, the 2-manifold that we use for experimentation is always S
2
. The filtration is generated by a manifold sweep, as described in Section 2.5.
Therefore, each filtration has length equivalent to the number of vertices in the manifold, which is one more than the size of the grid because of the dummy
vertex. For example, Sine has a filtration of 100 × 100 + 1 = 10, 001 com-
plexes.
12.5.2 Timings and Statistics
We first compute a filtration of the sphere triangulation by a manifold sweep. We then use the persistence algorithm to compute and classify the critical
Fig. 12.9. The data sets in Table 12.14 rendered as gray-scale images. The intensity of each pixel of the image corresponds to the relative height at that location.
Table 12.15. The number of critical points of the four triangulated spheres. The Mon column gives the number of 2-fold monkey saddles. Note that
Min − Sad − 2Mon + Max = 2 in each case, as it should be.
Min Sad
Mon Max
Sine 10
24 16
Iran 1,302
2,786 27
1,540 Himalayas
2,132 4,452
51 2,424
Andes 20,855
38,326 1,820
21,113 North America
15,032 30,733
464 16,631
points using the procedure described in Section 6.2.3. Table 12.15 lists the number of critical points of each type. As we start with grid data and add di-
agonals in a consistent manner, each vertex other than the dummy vertex has degree 6. Therefore, monkey saddles are the only multiple saddles that may
